Integrand size = 19, antiderivative size = 849 \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^3} \, dx=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
1/4*x*(e*x+d)^(1/2)/a/(c*x^2+a)^2+1/16*(a*d*e+(5*a*e^2+6*c*d^2)*x)*(e*x+d) ^(1/2)/a^2/(a*e^2+c*d^2)/(c*x^2+a)+1/64*e*arctanh((-c^(1/4)*2^(1/2)*(e*x+d )^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1 /2))^(1/2))*(6*c^(3/2)*d^3+8*a*d*e^2*c^(1/2)+(5*a*e^2+6*c*d^2)*(a*e^2+c*d^ 2)^(1/2))/a^2/c^(3/4)/(a*e^2+c*d^2)^(3/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2) ^(1/2))^(1/2)-1/64*e*arctanh((c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a* e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(6*c^(3/2) *d^3+8*a*d*e^2*c^(1/2)+(5*a*e^2+6*c*d^2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(3/4)/ (a*e^2+c*d^2)^(3/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/128*e* ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^ (1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(6*c^(3/2)*d^3+8*a*d*e^2*c^(1/2)-(5*a*e^ 2+6*c*d^2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(3/4)/(a*e^2+c*d^2)^(3/2)*2^(1/2)/(d *c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)+1/128*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^ 2)^(1/2)+c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/ 2))*(6*c^(3/2)*d^3+8*a*d*e^2*c^(1/2)-(5*a*e^2+6*c*d^2)*(a*e^2+c*d^2)^(1/2) )/a^2/c^(3/4)/(a*e^2+c*d^2)^(3/2)*2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^ (1/2)
Result contains complex when optimal does not.
Time = 1.85 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {d+e x} \left (6 c^2 d^2 x^3+a^2 e (d+9 e x)+a c x \left (10 d^2+d e x+5 e^2 x^2\right )\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {i \left (12 c d^2+18 i \sqrt {a} \sqrt {c} d e-5 a e^2\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {c} \left (\sqrt {c} d+i \sqrt {a} e\right ) \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {i \left (12 c d^2-18 i \sqrt {a} \sqrt {c} d e-5 a e^2\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {c} \left (\sqrt {c} d-i \sqrt {a} e\right ) \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{32 a^{5/2}} \]
((2*Sqrt[a]*Sqrt[d + e*x]*(6*c^2*d^2*x^3 + a^2*e*(d + 9*e*x) + a*c*x*(10*d ^2 + d*e*x + 5*e^2*x^2)))/((c*d^2 + a*e^2)*(a + c*x^2)^2) + (I*(12*c*d^2 + (18*I)*Sqrt[a]*Sqrt[c]*d*e - 5*a*e^2)*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqr t[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/(Sqrt[c]*(Sqrt[c]*d + I *Sqrt[a]*e)*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]) - (I*(12*c*d^2 - (18*I)*Sq rt[a]*Sqrt[c]*d*e - 5*a*e^2)*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sq rt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/(Sqrt[c]*(Sqrt[c]*d - I*Sqrt[a]*e )*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]))/(32*a^(5/2))
Time = 1.41 (sec) , antiderivative size = 909, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {494, 27, 686, 27, 654, 27, 1483, 27, 1142, 25, 27, 1083, 219, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 494 |
| \(\displaystyle \frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}-\frac {\int -\frac {6 d+5 e x}{2 \sqrt {d+e x} \left (c x^2+a\right )^2}dx}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {6 d+5 e x}{\sqrt {d+e x} \left (c x^2+a\right )^2}dx}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (x \left (5 a e^2+6 c d^2\right )+a d e\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac {\int -\frac {c \left (d \left (12 c d^2+13 a e^2\right )+e \left (6 c d^2+5 a e^2\right ) x\right )}{2 \sqrt {d+e x} \left (c x^2+a\right )}dx}{2 a c \left (a e^2+c d^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {d \left (12 c d^2+13 a e^2\right )+e \left (6 c d^2+5 a e^2\right ) x}{\sqrt {d+e x} \left (c x^2+a\right )}dx}{4 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} \left (x \left (5 a e^2+6 c d^2\right )+a d e\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {\int \frac {e \left (2 d \left (3 c d^2+4 a e^2\right )+\left (6 c d^2+5 a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} \left (x \left (5 a e^2+6 c d^2\right )+a d e\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e \int \frac {2 d \left (3 c d^2+4 a e^2\right )+\left (6 c d^2+5 a e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} \left (x \left (5 a e^2+6 c d^2\right )+a d e\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\frac {e \left (\frac {\int \frac {2 \sqrt {2} d \left (3 c d^2+4 a e^2\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {2 \sqrt {2} d \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2+4 a e^2\right )+\sqrt [4]{c} \left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{2 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} \left (x \left (5 a e^2+6 c d^2\right )+a d e\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e \left (\frac {\int \frac {2 \sqrt {2} d \left (3 c d^2+4 a e^2\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {2 \sqrt {2} d \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2+4 a e^2\right )+\sqrt [4]{c} \left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{2 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} \left (x \left (5 a e^2+6 c d^2\right )+a d e\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\sqrt {d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac {\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{2 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )}+\frac {e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}-\frac {1}{2} \sqrt [4]{c} \left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {1}{2} \sqrt [4]{c} \left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a \left (c d^2+a e^2\right )}}{8 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac {\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{2 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )}+\frac {e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {1}{2} \sqrt [4]{c} \left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {1}{2} \sqrt [4]{c} \left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a \left (c d^2+a e^2\right )}}{8 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac {\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{2 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )}+\frac {e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {\left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {\left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a \left (c d^2+a e^2\right )}}{8 a}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\sqrt {d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac {\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{2 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )}+\frac {e \left (\frac {\frac {\left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {c}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {c}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a \left (c d^2+a e^2\right )}}{8 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac {\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{2 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )}+\frac {e \left (\frac {\frac {\left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a \left (c d^2+a e^2\right )}}{8 a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\sqrt {d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac {\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{2 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )}+\frac {e \left (\frac {-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {1}{2} \sqrt [4]{c} \left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {1}{2} \sqrt [4]{c} \left (6 c d^3+8 a e^2 d-\frac {\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )}{\sqrt {c}}\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a \left (c d^2+a e^2\right )}}{8 a}\) |
(x*Sqrt[d + e*x])/(4*a*(a + c*x^2)^2) + ((Sqrt[d + e*x]*(a*d*e + (6*c*d^2 + 5*a*e^2)*x))/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) + (e*((-((Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a *e^2]*(6*c*d^2 + 5*a*e^2))*ArcTanh[(c^(1/4)*(-((Sqrt[2]*Sqrt[Sqrt[c]*d + S qrt[c*d^2 + a*e^2]])/c^(1/4)) + 2*Sqrt[d + e*x]))/(Sqrt[2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])])/(c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])) - (c^(1/4)*(6*c*d^3 + 8*a*d*e^2 - (Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2 ))/Sqrt[c])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqr t[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/2)/(2*Sqrt[2]*Sqrt[c ]*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (-((Sqrt[Sq rt[c]*d + Sqrt[c*d^2 + a*e^2]]*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 + Sqrt[c *d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*ArcTanh[(c^(1/4)*((Sqrt[2]*Sqrt[Sqrt[c] *d + Sqrt[c*d^2 + a*e^2]])/c^(1/4) + 2*Sqrt[d + e*x]))/(Sqrt[2]*Sqrt[Sqrt[ c]*d - Sqrt[c*d^2 + a*e^2]])])/(c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^ 2]])) + (c^(1/4)*(6*c*d^3 + 8*a*d*e^2 - (Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5* a*e^2))/Sqrt[c])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/2)/(2*Sqrt[2]*S qrt[c]*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])))/(2*a*( c*d^2 + a*e^2)))/(8*a)
3.7.46.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-x)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^(p + 1)*(c*(2*p + 3) + d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 0] && (Lt Q[n, 1] || (ILtQ[n + 2*p + 3, 0] && NeQ[n, 2])) && IntQuadraticQ[a, 0, b, c , d, n, p, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Timed out.
\[\int \frac {\sqrt {e x +d}}{\left (c \,x^{2}+a \right )^{3}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 3770 vs. \(2 (695) = 1390\).
Time = 1.26 (sec) , antiderivative size = 3770, normalized size of antiderivative = 4.44 \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
1/64*((a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2* d^2 + a^4*c*e^2)*x^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d ^3*e^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2* e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^1 4)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^ 6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c ^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))*log((3024*c^3 *d^6*e^5 + 7884*a*c^2*d^4*e^7 + 5625*a^2*c*d^2*e^9 + 625*a^3*e^11)*sqrt(e* x + d) + (126*a^3*c^3*d^5*e^6 + 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^10 - ( 12*a^5*c^7*d^10 + 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 + 84*a^8*c^4*d^4 *e^6 + 34*a^9*c^3*d^2*e^8 + 5*a^10*c^2*e^10)*sqrt(-(441*c^2*d^4*e^10 + 105 0*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7 *c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^ 10 + a^11*c^3*e^12)))*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d ^3*e^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2* e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^1 4)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^ 6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c ^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))) - (a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2*d^2 + a^4*c*e...
Timed out. \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^3} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + a\right )}^{3}} \,d x } \]
Time = 0.47 (sec) , antiderivative size = 1068, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^3} \, dx=-\frac {{\left ({\left (a^{2} c d^{2} e + a^{3} e^{3}\right )}^{2} {\left (6 \, c d^{2} e + 5 \, a e^{3}\right )} {\left | c \right |} - 2 \, {\left (3 \, \sqrt {-a c} a c^{2} d^{5} e + 7 \, \sqrt {-a c} a^{2} c d^{3} e^{3} + 4 \, \sqrt {-a c} a^{3} d e^{5}\right )} {\left | -a^{2} c d^{2} e - a^{3} e^{3} \right |} {\left | c \right |} + {\left (12 \, a^{3} c^{4} d^{8} e + 37 \, a^{4} c^{3} d^{6} e^{3} + 38 \, a^{5} c^{2} d^{4} e^{5} + 13 \, a^{6} c d^{2} e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{2} d^{3} + a^{3} c d e^{2} + \sqrt {{\left (a^{2} c^{2} d^{3} + a^{3} c d e^{2}\right )}^{2} - {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )}}}{a^{2} c^{2} d^{2} + a^{3} c e^{2}}}}\right )}{32 \, {\left (a^{4} c^{3} d^{4} e + 2 \, a^{5} c^{2} d^{2} e^{3} + a^{6} c e^{5} - \sqrt {-a c} a^{3} c^{3} d^{5} - 2 \, \sqrt {-a c} a^{4} c^{2} d^{3} e^{2} - \sqrt {-a c} a^{5} c d e^{4}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | -a^{2} c d^{2} e - a^{3} e^{3} \right |}} - \frac {{\left ({\left (a^{2} c d^{2} e + a^{3} e^{3}\right )}^{2} {\left (6 \, c d^{2} e + 5 \, a e^{3}\right )} {\left | c \right |} + 2 \, {\left (3 \, \sqrt {-a c} a c^{2} d^{5} e + 7 \, \sqrt {-a c} a^{2} c d^{3} e^{3} + 4 \, \sqrt {-a c} a^{3} d e^{5}\right )} {\left | -a^{2} c d^{2} e - a^{3} e^{3} \right |} {\left | c \right |} + {\left (12 \, a^{3} c^{4} d^{8} e + 37 \, a^{4} c^{3} d^{6} e^{3} + 38 \, a^{5} c^{2} d^{4} e^{5} + 13 \, a^{6} c d^{2} e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{2} d^{3} + a^{3} c d e^{2} - \sqrt {{\left (a^{2} c^{2} d^{3} + a^{3} c d e^{2}\right )}^{2} - {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )}}}{a^{2} c^{2} d^{2} + a^{3} c e^{2}}}}\right )}{32 \, {\left (a^{4} c^{3} d^{4} e + 2 \, a^{5} c^{2} d^{2} e^{3} + a^{6} c e^{5} + \sqrt {-a c} a^{3} c^{3} d^{5} + 2 \, \sqrt {-a c} a^{4} c^{2} d^{3} e^{2} + \sqrt {-a c} a^{5} c d e^{4}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | -a^{2} c d^{2} e - a^{3} e^{3} \right |}} + \frac {6 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} d^{2} e - 18 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d^{3} e + 18 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{4} e - 6 \, \sqrt {e x + d} c^{2} d^{5} e + 5 \, {\left (e x + d\right )}^{\frac {7}{2}} a c e^{3} - 14 \, {\left (e x + d\right )}^{\frac {5}{2}} a c d e^{3} + 23 \, {\left (e x + d\right )}^{\frac {3}{2}} a c d^{2} e^{3} - 14 \, \sqrt {e x + d} a c d^{3} e^{3} + 9 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} e^{5} - 8 \, \sqrt {e x + d} a^{2} d e^{5}}{16 \, {\left (a^{2} c d^{2} + a^{3} e^{2}\right )} {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + a e^{2}\right )}^{2}} \]
-1/32*((a^2*c*d^2*e + a^3*e^3)^2*(6*c*d^2*e + 5*a*e^3)*abs(c) - 2*(3*sqrt( -a*c)*a*c^2*d^5*e + 7*sqrt(-a*c)*a^2*c*d^3*e^3 + 4*sqrt(-a*c)*a^3*d*e^5)*a bs(-a^2*c*d^2*e - a^3*e^3)*abs(c) + (12*a^3*c^4*d^8*e + 37*a^4*c^3*d^6*e^3 + 38*a^5*c^2*d^4*e^5 + 13*a^6*c*d^2*e^7)*abs(c))*arctan(sqrt(e*x + d)/sqr t(-(a^2*c^2*d^3 + a^3*c*d*e^2 + sqrt((a^2*c^2*d^3 + a^3*c*d*e^2)^2 - (a^2* c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4)*(a^2*c^2*d^2 + a^3*c*e^2)))/(a^2*c^2* d^2 + a^3*c*e^2)))/((a^4*c^3*d^4*e + 2*a^5*c^2*d^2*e^3 + a^6*c*e^5 - sqrt( -a*c)*a^3*c^3*d^5 - 2*sqrt(-a*c)*a^4*c^2*d^3*e^2 - sqrt(-a*c)*a^5*c*d*e^4) *sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(-a^2*c*d^2*e - a^3*e^3)) - 1/32*((a^2*c *d^2*e + a^3*e^3)^2*(6*c*d^2*e + 5*a*e^3)*abs(c) + 2*(3*sqrt(-a*c)*a*c^2*d ^5*e + 7*sqrt(-a*c)*a^2*c*d^3*e^3 + 4*sqrt(-a*c)*a^3*d*e^5)*abs(-a^2*c*d^2 *e - a^3*e^3)*abs(c) + (12*a^3*c^4*d^8*e + 37*a^4*c^3*d^6*e^3 + 38*a^5*c^2 *d^4*e^5 + 13*a^6*c*d^2*e^7)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a^2*c^2*d ^3 + a^3*c*d*e^2 - sqrt((a^2*c^2*d^3 + a^3*c*d*e^2)^2 - (a^2*c^2*d^4 + 2*a ^3*c*d^2*e^2 + a^4*e^4)*(a^2*c^2*d^2 + a^3*c*e^2)))/(a^2*c^2*d^2 + a^3*c*e ^2)))/((a^4*c^3*d^4*e + 2*a^5*c^2*d^2*e^3 + a^6*c*e^5 + sqrt(-a*c)*a^3*c^3 *d^5 + 2*sqrt(-a*c)*a^4*c^2*d^3*e^2 + sqrt(-a*c)*a^5*c*d*e^4)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*abs(-a^2*c*d^2*e - a^3*e^3)) + 1/16*(6*(e*x + d)^(7/2)*c ^2*d^2*e - 18*(e*x + d)^(5/2)*c^2*d^3*e + 18*(e*x + d)^(3/2)*c^2*d^4*e - 6 *sqrt(e*x + d)*c^2*d^5*e + 5*(e*x + d)^(7/2)*a*c*e^3 - 14*(e*x + d)^(5/...
Time = 12.08 (sec) , antiderivative size = 6238, normalized size of antiderivative = 7.35 \[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
atan(((((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 + 57344*a^6*c^4*d^3*e ^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 + 2*a^7*c*d^2*e^2)) - ((d + e*x)^(1/2)*(4 096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 + 8192*a^6*c^5*d^3*e^4)*(-(144*a^ 5*c^5*d^7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d ^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10 *c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) )/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^5*c^5*d^7 - 25* a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^ 7*c^3*d^3*e^4 - 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13 *c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) - ((d + e*x)^( 1/2)*(144*c^5*d^6*e^2 - 25*a^3*c^2*e^8 + 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d ^2*e^6))/(64*(a^6*e^4 + a^4*c^2*d^4 + 2*a^5*c*d^2*e^2)))*(-(144*a^5*c^5*d^ 7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 + a^13*c^3*e^6 + 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2)*1i - (( (32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 + 57344*a^6*c^4*d^3*e^5)/(409 6*(a^8*e^4 + a^6*c^2*d^4 + 2*a^7*c*d^2*e^2)) + ((d + e*x)^(1/2)*(4096*a^7* c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 + 8192*a^6*c^5*d^3*e^4)*(-(144*a^5*c^5*d^ 7 - 25*a*e^7*(-a^15*c^3)^(1/2) + 105*a^8*c^2*d*e^6 + 420*a^6*c^4*d^5*e^2 + 385*a^7*c^3*d^3*e^4 - 21*c*d^2*e^5*(-a^15*c^3)^(1/2))/(4096*(a^10*c^6*...